Winning Wordle
github.com
github.com
> Q: What simple strategy guarantees a win within 6 guesses?
I've seen a lot of analyses of Wordle that pose this question, and I feel like these are not people who play Wordle (not unlikely - these are seemingly people more interested in analysing the game from a statistical standpoint than simply playing directly).
But while the answer to this question is interesting enough, it's not that interesting. People I've spoken to who play have zero interest in winning in 5 or 6 guesses - in fact most would consider a score of 6/6 a loss (or at least a relatively bad score), unless it was a particularly tricky word.
> (1) If I win in two guesses, am I good or lucky?
Q: What first guess maximizes the number of guaranteed wins on the second guess?
Q: What first guess maximizes the number of expected wins on the second guess?
> The probability of winning in two guesses is about 2%, so the answer is: mostly lucky.
These - and the follow-ups I was hoping for for 3, 4 & 5 guesses respectively - are much more interesting questions.
* Yes I can just clone this and edit to find these things out (& probably will later this week) but I'm just commenting on my confusion around why there's so much focus on the least interesting part of a problem.
Interesting thought. But idunno, I am prettty into wordle, I agree with you, my goal with the daily is to get it in 3 as often as possible, but when it comes to "extracurriculars" I'm interested in all sorts of questions that are divorced with this concern. And I'm actually not that interested in deeply analyzing the part I enjoy most (getting the daily in 3).
> I'm actually not that interested in deeply analyzing the part I enjoy most (getting the daily in 3).
I think I'm similar; I've been using the same starter word since almost day 1, and despite having seen lots of very compelling evidence that it is suboptimal, I have not changed it :D
But I'm still somehow interested in what is optimal (& I guess oddly passionate about the definition of optimality).
BLA*E
Where I try
BLADE
BLAZE
BLARE
BLAME
Those are frustrating, but not my fault (assuming I don't re-use any letters).
A guess like DREAM would guarantee the next guess is correct.
It's in an important sense actually easier! Sure, good play results in more guesses than in not-hard-mode, but that would be like saying a 100m sprint is easier than a 2-mile race because the latter takes longer. The more interesting measure of difficulty is how hard it is play optimally. And since the search space for not-hard-mode is bigger, I'm pretty sure it's harder.
But more importantly, hard mode is also less fun, and this thread demonstrates exactly why: the creative aspect is coming up with le mot juste for quickly honing in on the right answer. Hard mode takes away your tools to do that and sometimes even results in a thoughtless guessing game.
'blade', 'blake', 'blame', 'blare', 'blase', 'blate', 'blaze'
wordle tends to be more familiar words, so I'd ignore blate and blake. So now we have D, M, R, S, Z
Doesn't look like there's a word that has 4 of these letters to guarantee on next guess. So need to try for the most likely 3 letters. Could be 2 more guesses, but still ends up better than a 1 in 5 chance.
I pulled the wordle word list from the page source of the game.
This, again, depends what your goal is, or what you consider "winning". If you're only focused on avoiding losing, then yes this is the right move. If you're trying for a low score, it's not as clear, and hard mode may actually be "easier" in many ways.
If you want to maximize your number of three-guess-wins even if your 6-guess-failure rate is higher, it is obviously can't be a correct strategy to use your third guess on a word that you know can't be correct.
Maximizing information from each guess until the answer is down to two candidates minimizes the number of guesses needed to win.
If someone considers a 4 to be a soft-failure then they will prefer (3,6,6) to (4,4,4) even though the latter is a lower mean number of guesses: it's a higher count of 4+ 'failures'. The strategy that considers (3,6,6) a better result than (4,4,4) wouldn't guess "dream" in the above example.
Reading that wiki page brings up a follow-up question:
Given:
- the optimal starter word appears to be the quite obtuse word "SALET"
- while we don't know that "SALET" won't someday be an answer, there's a common assumption among players that the answer list consists of "common" words. If I'm not mistaken this is a founded assumption
What would the rankings of the subset of optimal starter words that are "common" be (i.e. would be likely to be reasonable guesses of typical players)?
[0] https://www.poirrier.ca/notes/wordle-optimal/
[1] http://sonorouschocolate.com/notes/index.php?title=The_best_...
> I definitely have not used the "optimal strategy" every time, if I even knew that strategy in the first place.
It's also worth noting that these models (presumably) assume an equal chance for every word in the provided dictionary, whereas - assuming the answers are "common" words - a human will likely be strongly biased toward the subset of possible answers and therefore can have much higher probabilities of success in practice than the theoretically-optimal strategies.
Yes, we do ... all the answers and the day they will occur are known.
That's for any given game, so presumably if your statistics over time are significantly higher than 2% then the answer is mostly good? Or perhaps that's a statistical fallacy akin to tossing a head 5 times in a row?
It's a game that you play for fun. If folks want to analyze the dictionaries to develop their strategy because it's fun for them, then they're doing it right. Not cheating.
If you prize Wordle solution patterns that are symmetrical or aesthetically pleasing because it's fun, then you're doing it right.
If you try to see how many consecutive hard-mode guesses you can make with NO correct letters because it's fun, then you're doing it right.
If you try to see how far you can skew your distribution towards six without ever losing because it's fun, then you're doing it right.
I played the game maybe twice, the game itself is just very boring and hard for me (though generally, I hate playing games, so it's not a critique on the game).
But!
I read the source code and checked the implementation. I read many different strategies optimizing for different aspects of the game. I implemented a very naive solver in JavaScript, then I thought about more advanced data structures and algorithms I could use to improve my naive solver (both from the game's POV and from the computational POV). I watched videos cloning the app with different frameworks (or without) and technologies (mobile, command line, web) and I listened to the author on a podcast.
I'm having fun and I'm learning while doing these things, so I appreciate Wordle, even if I can't be bothered to solve the daily word.
I made a browser extension to show remaining solution counts and make it easier to restart the game without losing my streak - I usually use it to play around more after solving the day's word.
Bringing this down to 5 would mean that one could always win at Wordle with the same set of 5 guesses. Seems unlikely that such a solution exists, but interesting question nonetheless.
> March 2nd, 2022: Armavica has found a set of 7 words that solve Wordle! They are CLANG FATTY ODDER RUMBA SKILL VERGE WHOOP.
I once got it on second word - it was pure chance. I am now waiting for the day when I get it on the first attempt. :-)
But a human player though might do better by priming their pattern matcher with something more likely to trigger good hints, such as knowing more vowels. My start word is often AEONS
SOARE
CLINT
PUDGY
Developed it looking at the answer and guess lists and finding words that maximize frequency and placement. E.g. Y and E are very commonly seen in fifth position. No code, just excel and searching the word lists with regular expressions.
It's so effective the game lost a lot of its fun :( I almost always get the answer on guess #4. The days I struggle and get it on guess 5 or 6 are fun though! :)
PARTY
has also been so effective for me it almost takes the fun out of it. Usually get it in 3/6, but never more than 4.
The first two to get all the verbs in and then NYMPH to get as many consonants as possible, it's generally not too bad it seems.
If there are no S's I'll start the next word with a CH, if there is an S in first place I'll start with SH. Pretty consistent 3-4.
That is, you can have fun with the analysis and with playing the game as a human.
I'm interested to know how things change if you require hard mode.
That is, the math is clearly accurate and describes how you can do with perfect knowledge. If you want to grade yourself against that, by all means consider it. For casual play, though, these numbers are not at all a reflection on what your rates will be.
1. Guessing the first word(s) randomly, and then solving from there. Not cheating at all, but frustrating for me, as I am terrible at anagrams.
2. Using a generic dictionary to check possible answers against. Not cheating. I use the standard dictionary in my Linux distro, filtered by 5-letter words.
3. Using the dictionary from the source code to check against. Skirting dangerously close to cheating, but probably OK.
4. Using the list of possible answers. Clearly cheating.
5. Using starting guesses that I concocted myself by studying a generic letter frequency table. Not cheating.
6. Using optimal starting guesses, generated by someone else, based on the WORDLE letter frequencies. Cheating.
7. Using software that I wrote to automate my elimination process based on the dictionary in (2). Again, skirting close to cheating, but possibly OK.
8. Using someone else's software to guess, based on solution graphs. Cheating.
I guess I'll try this notebook tomorrow and see how it rates my opening.
BRING
CHUMP
DOWLY
The first 1-3 words may "strike" early. But victory is inevitable it seems with those. They seem pretty durable against sneaky double-letter words too.
HANDY SWIFT GLOVE CRUMP seem to lack the concentration of "quick strike" letter combos onthe first two words (FEAST and BRING for me). Then CHUMP and DOWLY are "cleanup" guesses if not enough comes out of FEAST / BRING.
But I don't have math behind it so... what do I really know.
1 - 0 (0%)
2 - 7 (17%)
3 - 13 (32%)
4 - 16 (39%)
5 - 4 (10%)
6 - 1 (2%)
lose - 0 (0%)
My strategy:
- Always start with the same word, one that has the most common letters
- Prefer more common letters to less common letters
- Prefer more common words to less common words
- Starting with round three, never guess until I know all possible common words
- Don't repeat a letter unless the only other possible words are uncommon
- Prefer consonant combinations at the start of the word over staring with a vowel
- Never do plurals (I think the game does not use them)
A guiding principle is maximize learning at each round. I have a manual written process for listing all possible combinations at each step and iterating on each to find all possible words. I usually start that on the third round.
I've written a small program that takes a pattern (like ??EA?) and a list of eliminated letters and spits out all possible words. My wife says it's cheating, so I rarely use it.
I don't think it's cheating because it only saves me time going through all possible combinations on paper. I guess there's some chance it will suggest a common word I would have never hit on manually.
I don't deny that my high rate of second guess wins is partly luck. But I contend that the N is large enough by now that it's at least partly my strategy.
I always apply all learnings to the next guess. That is, with each guess I use all the confirmed letters and put them in the confirmed positions. This DOES NOT maximize learning. That would require using all new letters every time until you know the word for sure. But this would reduce overall performance by eliminating luck in early rounds. So it's more correct to say my strategy balances between learning and serendipity.
But the analysis only uses the latter subset. So in theory there could be a valid 5-letter that could be a good opener, even if it cannot be a winning word.
It removes a lot of the fog of war and since it leaves me with few guesses remaining, keeps the game tense enough to be fun
Are you feeling ok?
The fun comes from pretending that's not the case and playing it straight. And then from being able to use the oracle as an aid to analysis, if you're into that sort of thing.
I used to start with TEARS. Once I realized that they never used plurals, I switched to STARE, since putting the S at the end doesn't appreciably lower the entropy. I usually end up winning with 3 guesses, which is the point where strategy starts to pay off. (I do agree with Peter that 2-guess wins are almost always due to luck.)
In other words, Norvig's approach requires you to guess letters that aren't that likely to appear at all. He will get more 2-guess wins, but I will do better in the long run.
You seem not to have read past the first few lines of the article.
Norvig's Sudoku solver inspired me to make my own Wordle solver last month
Edit:. Ah, I was reversing which was the guess and which was the answer. Makes sense now.
FLOSS
TEETH
>What first guess maximizes the number of expected wins on the second guess? A: FILET gives you 57.5 expected wins (out of 2,315)
Note that 57.5 / 2315 = 2.4%, so being "good" at the first word can only improve your chances of a two-guess win by at most ~20%. Not that much.
I believe that stat illuminates a MAJOR factor for why Wordle is so popular that I haven't seen discussed yet: it's a game of (almost) pure luck, that successfully gives every man, woman and child of diverse intelligence the perception of being very smart, occasionally. It's a pat-on-the-back generator that works for everyone, not just those with high aptitude.
It's a lottery ticket, if winning the lottery meant you could show off how smart you are.
That said, it's probably easier than it looks at first glance and given a decent vocabulary and some sense for word and word pair frequencies in different positions you'll probably finish at 6 turns or less unless you really go down a bad path.